Abstract: This talk consists of two parts. The first part is almost entirely devoted to a discussion of Hilbert’s finitist metamathematics in the 1920s, with particular emphasis on his conception of finitist consistency proofs for formalized mathematical theories T. When Hilbert wrote his famous essay ‘On the Infinite’ (1925-1926), his proof theory of the 1920s had grown to full maturity. It is here that he pays special attention to describing what is usually called “the method of ideal elements”, such as the postulation of points and lines at infinity in projective geometry or the postulation of the existence of n roots for an n-th degree polynomial in algebra. In subsequent smaller sections, I try to shed light on some difficulties to which Hilbert’s metamathematics of the 1920s gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Another difficulty emerges from Hilbert’s illicit assumptions of infinity in metamathematics. On the way, I shall comment on the relationship between finitism and intuitionism, on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn as well as on partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson.

In the second part of my talk, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929-1933. I argue that his insouciant attitude towards the emergence of a contradiction in a mathematical calculus as well as his outright rejection of metamathematical consistency proofs are unjustified. In particular, I argue — by way of presenting an imaginary dialogue between Wittgenstein and Hilbert — that Wittgenstein falls short of making a convincing case against Hilbert’s proof-theoretic project. I conclude with philosophical remarks on consistency proofs and the notion of soundness.